Let $\Delta$ be a Compact Hausdorff space in $\mathbb{C^n}$. Let $A$ be a closed sub algebra of $C(\Delta)$(space of all complex valued continuous functions on $\Delta$) which contains the constant functions. Let $\mathbb{D}$ be the open unit ball in the complex plane.

Let $A{(\mathbb{D})}=\{f\in C(\Delta): f(\Delta)\subset\mathbb{D} \}$ and $A({\bar{\mathbb{D}}})=\{f\in C(\Delta): f(\Delta)\subset\bar{\mathbb{D}} \}$.

Now let $a,b\in\Delta$, and consider the supremum $\sup_f\left\{\left|\frac{f(w_1)-f(w_2)}{1-\overline{f(w_2)}f(w_1)}\right|: f\in A{(\mathbb{D})} \right\}$. It is said that the above defined supremum is the same if we take $A({\bar{\mathbb{D}}})$ or $A{(\mathbb{D})}$. Can anyone tell how?